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Thermoelectric (TE) nanostructures with dimensions of 100 nm can show substantially better TE properties compared to the same material in the bulk form due to charge and heat transport effects specific to the nanometer scale. However, TE physics in nanostructures is still described using the Kelvin relation (KR) P = aT, where P is the Peltier coefficient, a the thermopower, and T the absolute temperature, even though derivation of the KR uses a local equilibrium assumption (LEA) applicable to macroscopic systems. It is unclear whether nanostructures with nanostructures with dimensions on the order of an inelastic mean free path satisfy a LEA under any nonzero temperature gradient. Here, we present an experimental test of the KR on a TE system consisting of doped silicon-based nanostructures with dimensions comparable to the phonon–phonon and electron–phonon mean-free-paths. Such nanostructures are small enough that true local thermodynamic equilibrium may not exist when a thermal gradient is applied. The KR is tested by measuring the ratio P/a under various applied temperature differences and comparing it to the average T. Results show relative deviations from the KR of |(P/a)/T –1| ≤ 2.2%, within measurement uncertainty. This suggests that a complete local equilibrium among all degrees of freedom may be unnecessary for the KR to be valid but could be replaced by a weaker condition of local equilibrium among only charge carriers. 

The Kelvin relation (KR) connecting the Peltier coefficient Π, the thermopower α, and the absolute temperature T via Π = αT is a cornerstone of thermoelectric (TE) physics. It is also a widely recognized example of an Onsager reciprocal relation, a foundational principle in nonequilibrium irreversible thermodynamics. While the KR is routinely invoked to understand TE systems, it has surprisingly little rigorous empirical verification. Accurate experimental tests of the KR are complicated by several factors, including non-Peltier heat flows such as Joule heating or Fourier thermal conduction, uncharacterized thermal contact impedances, and the need for Peltier and thermopower effects to be measured on the same thermopile at the same temperatures. Most empirical assessments of the KR have either made questionable simplifications or been limited in accuracy to several percent. Here, we present a test of the KR that is free of the difficulties of prior experiments and relies only on conventional voltage, current, and temperature measurements, so that it could be performed on any thermopile. Conducting the test on a Bi 2 Te 3 thermopile, the empirical ratio Π/α is found to equal T within a relative deviation < 0.5% for T in the range of 320–340 K. This result is quantitatively consistent with the KR and justifies the use of the KR in TE applications to reasonably high accuracy. 

Thermoelectric (TE) generators and coolers are one possible solution to energy autonomy for internet-of-things and biomedical electronics and to locally cool high-performance integrated circuits. The development of TE technology requires not only research into TE materials but also advancing TE device physics, which involves determining properties such as the thermopower ( α) and Peltier ( Π) coefficients at the device rather than material level. Although Π governs TE cooler operation, it is rarely measured because of difficulties isolating Π from larger non-Peltier heat effects such as Joule heating and Fourier thermal conduction. Instead, Π is almost always inferred from α via a theoretical Kelvin relation Π =  αT, where T is the absolute temperature. Here, we demonstrate a method for independently measuring Π on any TE device via the difference in heat flows between the thermopile held open-circuit vs short-circuit. This method determines Π solely from conventionally measured device performance parameters, corrects for non-Peltier heat effects, does not require separate knowledge of material property values, and does not assume the Kelvin relation. A measurement of Π is demonstrated on a commercial Bi 2 Te 3 TE generator. By measuring α and Π independently on the same device, the ratio ( Π/ α) is free of parasitic thermal impedances, allowing the Kelvin relation to be empirically verified to reasonable accuracy.